The prior art comprises Italian patent IT 1.246.168, which teaches a machine for fitting and removing tires which includes a self-centering platform, rotatable about a horizontal axis, and a tool-bearing tubular arm having an axis located, when in the work position, in a more or less perpendicular direction to an axis of rotation of the platform. At its base the arm is hinged and can slide axially along a parallel direction to the rotation axis of the platform. The arm bears, coaxially, at its upper part, a tool-bearing shaft on which two tools are arranged. The shaft is rotatable about its own longitudinal axis, enabling a single tool to be placed in the working position. The arm can act on both sides of the tire belonging to the wheel placed on the self-centering platform, thanks to the freedom to rotate the tools predisposed on the end of the tool-bearing arm by 180.degree.. By performing this rotation the tools can be switched to operate on the opposite side of the tire. The tool-bearing arm is made so that when the arm is moved about the axis to which it is inferiorly hinged, the tool housed there is repositioned. The total operations needed for fitting and removing a tire using the above-described prior-art machine are relatively long and laborious.